Cats breed fast. Sexual maturity is 6-9 months, pregnancy takes 2-3 months, and a mother cat can go into heat a month after giving birth. Litters are mostly 2-6 kittens, and there is no such thing as "cat menopause" - your ancient housecat well past the end of a wild cat's lifespan can still get pregnant.

As long as there's a few breeding pairs out there, they can repopulate quickly. You would have to go to a very deliberate effort to make them extinct. Or accidentally destroy everything they would eat - which, given that they will eat pretty much any animal they can take down, would be a serious global extinction event.